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Sturm-Liouville problems


orthogonality

• the most important property of eigenfunction of Sturm-Liouville problems is orthogonality.
• orthogonal definition

funtions

• orthogonal function 중 communication theory 분야에서 많이 이용되는 function 에는 sinusoidal function (advanced: exponential family with Euler's theory), sinc function 등이 있다.
• the Fourier series ^[각주:1] is a particular type of orthogonal series that is very useful in solving engineering problems, especially commnication problem. the orthogonal functions that are used are either sinusoids or, equivalently, complex exponential functions. mathematicians generally call any orthogonal series a Fourier series. (text: digital and analog communication system, p68)

     * reference: 강의노트, 강의정리: Comm theory and System: intro, basic concepts

Fourier analysis



Fourier series

• periodic function
• a function
  
   is called a periodic function if
  
   is defined for all real x (perhaps except at some points, like
  
   and if there is some positive number p, called a period of
  
  , such that

• first few case, we will study various functions
  
   of period
  
  .

• given function of period
  
  , Fourier series
• given function of period
  
   can be represented by a trigonometric series; Fourier series,

• coefficients of Fourier series are the so-called Fourier coefficients of
  
  , given by the Euler formulas

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Fourier series expansion and function approximation

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lecture note. Partial fraction expansion and Fourier series expansion

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Partial fraction expansion
Fourier series expansion
     periodic function (definition and example)
     Fourier series representation of periodic function, f(x)
     Fourier series coefficient: Euler's formula when period is 2*pi and p

lecture note. function approximation and the square error of function, frequency and amplitude

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fourier series expansion of f(x)
approximation of f(x), f_N(x)
the square error of f_N(x), E_N: definition, derivation formula which in practrical using
limitation of the square error of f_N(x), E*: definition, parseval's equality
frequency and amplitude spectrum, derivation other form of fourier series
     frequency spectrum of f(x)
     nth harmonic frequency of f(x)
     amplitude spectrum of f(x)
     nth harmonic of f(x)

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